Practice Maths

L14 — Financial Applications

Key Terms

Present value (P)
The amount to invest today to reach a target future sum; P = A ÷ (1 + r)n.
Future value (A)
The total value of an investment or debt after n years.
Inflation
The general rise in prices over time; C = C0(1 + i)n. Erodes purchasing power even when the dollar amount stays the same.
Loan / credit
Borrowed money that accumulates compound interest until repaid. Monthly compounding: A = P(1 + r/12)12n.
Multi-phase investment
An investment split across two or more periods at different rates. Apply the formula in sequence, using each period’s output as the next period’s principal.
Purchasing power
What money can actually buy — eroded by inflation over time.

Choosing the Right Formula

SituationModelFormula
Fixed dollar gain/loss per periodSimple interest / SL depreciationA = P(1 + rn) or V = P − Dn
Percentage gain per periodCompound interest / AppreciationA = P(1 + r)n
Percentage loss per periodReducing balance depreciationV = P(1 − r)n
Compounding k times per yearCompound (multiple periods)A = P(1 + r/k)kn
Find starting amount, given finalPresent valueP = A ÷ (1 + r)n
Prices rising over timeInflationC = C0(1 + i)n

Present Value

Sometimes you know the future amount and need to find the present amount. Rearrange the compound interest formula:

P = A ÷ (1 + r)n

This answers: “How much must I invest today to have $A after n years at rate r?”

Key Comparison

  • Simple interest beats compound interest in the short term when the simple rate > compound rate.
  • Compound interest always wins in the long term — exponential growth eventually outpaces linear growth.
  • Reducing balance depreciation always gives a higher book value than straight-line after the first year (for the same rate).
Hot Tip: To choose the right formula, ask two questions: (1) Is the value growing (+r) or shrinking (−r)? (2) Is the change a fixed dollar amount (linear, use ×rn) or a fixed percentage (exponential, use n)? Answering both questions picks the formula automatically.

Present Value

Present value (PV) tells you what a future sum is worth in today’s money. If you want to have $A after n years at compound rate r, the amount to invest now is:

P = A ÷ (1 + r)n

Worked Example 1 — Present Value

You need $20000 in 6 years. A savings account earns 5% p.a. compound. How much should you deposit today?

P = 20000 ÷ (1.05)6 = 20000 ÷ 1.34010 ≈ $14924

Inflation

Inflation erodes purchasing power. If the annual inflation rate is i, the future cost of an item currently priced C0 is:

C = C0(1 + i)n

Worked Example 2 — Inflation

A university degree currently costs $30000. At 3% annual inflation, find its cost in 10 years.

C = 30000 × (1.03)10 = 30000 × 1.34392 ≈ $40318

Compound Interest on Loans

Loans also accumulate compound interest. A credit card balance of $B at annual rate r, compounded monthly, grows as:

A = B(1 + r/12)12n

Making no repayments allows the balance to grow rapidly — always pay more than the minimum!

Worked Example 3 — Multi-step Investment

Invest $5000 for 4 years at 6% p.a., then move the whole amount to an account paying 8% p.a. for 3 more years.

Phase 1: A4 = 5000 × (1.06)4 = 5000 × 1.26248 = $6312.38

Phase 2: A7 = 6312.38 × (1.08)3 = 6312.38 × 1.25971 ≈ $7952

  1. Identify the correct model. Fluency

    State which financial model applies to each situation and write the appropriate formula.

    • (a) A city’s population grows by 3% each year.
    • (b) A photocopier loses $800 in value every year.
    • (c) A credit card charges 18% p.a. on the outstanding balance, compounded monthly.
    • (d) A government bond pays $120 per year on a $2000 face value.

  2. Loan calculations. Fluency

    For each loan, find the amount owing after the given time if no repayments are made.

    • (a) $2000 credit card balance at 20% p.a., compounded monthly, after 1 year.
    • (b) $8000 personal loan at 12% p.a., compounded monthly, after 3 years.
    • (c) $15000 car loan at 9% p.a., compounded monthly, after 5 years.
    • (d) $25000 student loan at 4.5% p.a., compounded annually, after 10 years.

  3. Inflation calculations. Fluency

    • (a) A house costs $400000 today. At 3% annual inflation, find its cost in 5 years.
    • (b) A weekly grocery basket costs $150. At 4% annual inflation, find its cost in 10 years.
    • (c) A car cost $25000 in 2010. If inflation averaged 2.5% p.a., find its equivalent price in 2025.
    • (d) At 5% annual inflation, use trial and error (testing n = 14 and n = 15) to find approximately when prices double.

  4. Present value. Fluency

    • (a) How much must you invest today at 4% p.a. compound to have $10000 in 5 years?
    • (b) How much must you invest at 6% p.a. compound to have $50000 in 10 years?
    • (c) You need $15000 in 3 years. An account earns 5% p.a. compound. Find the required deposit today.
    • (d) Using simple interest at 5% p.a., find the principal that grows to $5000 after 4 years.

  5. Read from the graph: comparing three investment strategies. Understanding

    Three investors each start with $5000. Strategy X earns 10% p.a. simple interest (blue). Strategy Y earns 8% p.a. compound interest (orange). Strategy Z earns 6% p.a. compound interest (green).

    X Y Z $5000 $7000 $9000 $11000 0 2 4 6 8 10 Years (n) Amount ($)
    • (a) At year 4, which strategy gives the largest amount? Estimate the value for each.
    • (b) Between which two years does Strategy Y (compound 8%) overtake Strategy X (simple 10%)?
    • (c) Estimate the value of each strategy at year 10.
    • (d) You need to invest for 10 years. Which strategy do you recommend, and why?

  6. Savings goal. Understanding

    Amir wants to buy a $25000 car in 5 years and currently has $15000 to invest.

    • (a) At 8% p.a. compound interest, will he have enough after 5 years? Calculate and decide.
    • (b) If not, by how much does he fall short?
    • (c) Set up the equation 15000(1 + r)5 = 25000 and use trial and error (try r = 10% and 11%) to find the rate he needs.
    • (d) How much would he need to invest at 8% p.a. to guarantee $25000 after 5 years? (Use present value.)

  7. Credit card debt. Understanding

    Emma has a $3000 credit card balance at 24% p.a. compounded monthly.

    • (a) If she makes no payments, find her balance after 6 months.
    • (b) If she makes no payments, find her balance after 1 year.
    • (c) She pays $100 at the end of each month. At the start of month 1 she owes $3000; after one month’s interest she owes 3000×1.02 = $3060. After paying $100, her balance is $2960. Complete this process for months 2 and 3. Is she reducing the debt each month?
    • (d) Monthly interest on $3000 is $60. Her $100 payment covers $60 of interest and reduces the debt by $40. Approximately how many months will it take to clear the full $3000? (Use 3000 ÷ 40.)

  8. Real estate vs shares. Understanding

    $200000 is available to invest. Option R: property appreciating at 5% p.a. Option S: shares growing at 8% p.a.

    • (a) Find the value of each investment after 5 years.
    • (b) Find the value of each investment after 15 years.
    • (c) How much more is Option S worth than Option R after 15 years?
    • (d) In year 5, shares fall 30% due to a market crash, then recover at 8% p.a. for the remaining 10 years. Find the value of Option S at the end of 15 years under this scenario.

  9. Multi-phase investment. Problem Solving

    Jake invests $10000 at 7% p.a. compound for 5 years. He then moves the full amount to an account earning 9% p.a. compound for 3 more years.

    • (a) Find the value of his investment after the first 5 years.
    • (b) Find the value at the end of the full 8 years.
    • (c) How much total interest has he earned over the 8 years?
    • (d) Compare: if Jake had kept the money at 7% p.a. for all 8 years, what would he have? How much extra did switching to 9% earn him?

  10. Early loan repayment. Problem Solving

    Sarah takes out a $20000 personal loan at 8% p.a. compound interest (annual).

    • (a) If no repayments are made, find the balance owing after 3 years.
    • (b) Instead, Sarah makes a one-off repayment of $5000 at the end of year 1. Find her balance at the end of years 1, 2 and 3 (after the repayment is applied at the end of year 1).
    • (c) How much did the $5000 early payment save her (compared with part a)?
    • (d) Suppose instead of repaying, Sarah had invested the $5000 for 2 years at 8% p.a. Compare the result with your answer to (c). What does this suggest?
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